Elements of an operator calculus

Abstract

The operational calculus developed by the Irish School of the Vienna Development Method (VDM) has a life of its own independent of its applications. This thesis is interested in the operational calculus that arose from the modelling of information systems. One was not overly interested in the models themselves but in the operational calculus itself. The time one has invested in the analyses and development of the operational calculus has had the following results: (i) Refined and improved the operational calculus of the Irish School of the VDM. (ii) Altered the philosophy of the Irish School of the VDM by encouraging a shift away from a pure constructive approach that allowed the embracing of the totality of mathematics. (iii) Categorical semantics for partial map override in terms of topos theory. (iv) Refined the algebraic foundations of the operational calculus of the Irish School of the VDM with the result that operator identities that were not classified originally are now classified. The Irish School of the VDM has developed down many roads over the past seven years and one believes that one has contributed significantly to developing the operational calculus.